Reducing Rational Expressions by Factoring Polynomials

In an earlier section you learned about reducing rational expressions (also called Algebraic Fractions).  One of the key take-away ideas involved reducing.  Essentially, each term in the numerator and denominator must contain a common factor (multiplicative part).  That common factor can be divided out of all terms, essentially “un-distributed,” and then reduced to one.  

We apply that same concept with a new skill of factoring polynomials in this section.  Rational expressions can be daunting.  But, when you learn to break them down in your thinking and see key features and components for what they really are, and how they are related, they become much more manageable.  That’s our goal here!

Read through the notes below, taking notes on your own, and practicing the examples.  Watch the videos, then try the practice problem.  By working independently on this you are taking ownership of your learning.  That’s a powerful experience.  It might be difficult, but with time it gets easier and you’ll be better for the experience.

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In this section we will apply what we’ve learned about factoring polynomials to algebraic fractions, or rational expressions.  For example, we will learn to reduce something like that which follows by first factoring the polynomial in the numerator and denominator.

This expression above is NOT a polynomial because it has the variable in the denominator.  However, we can use what we’ve learned so far to simplify this.  The key idea in reducing is that each term must have a common factor.  This gets tricky when considering what a factor is. 

When you worked on these expressions previously, you only knew of constants, or monomials, as factors.  So the above example would be in fully reduced form because the three terms in the numerator do not share a common factor, nor do the three in the denominator, much less all six terms!  (Remember, a term is a part, separated by addition or subtraction.)

Let’s factor these expressions and see what we find.

Note: The numerator is simpler to factor than the denominator.  The simpler expression will often contain a clue to get you started in factoring the other, more difficult, expression.  After all, these are written to be reduced!  This is a great way to practice factoring because you’ll learn a new application for the skill.

Let’s take a closer look at what we have now. 

Here we have one term in the numerator and one term in the denominator.  The single term in each is comprised of a pair of binomials.  Remember, a term is separated by addition or subtraction.  These binomials are multiplying, not adding or subtracting.  To reduce, each tem must have a common factor.  A factor is a multiplicative part.  Since they both share the factor of x + 2, we can reduce by dividing each by x + 2.

Our final expression has two terms in the numerator, and two in the denominator.  Each is a binomial by itself.  While the first term of each has a factor for x, not ALL terms contain a common factor.

Key Ideas:

  1. Terms are parts separated by addition or subtraction
  2. To reduce a rational expression, each term must have a common factor.
  3. A factor is a multiplicative part

Here’s a summary of how to reduce these algebraic fractions (or rational expressions, they are the same thing by a different name):

  1. Factor the numerator and denominator completely.
  2. Look for common factors
  3. Reduce those factors out

As is the case when learning something new, the master of the new skill in the application of that skill is what is most difficult.  Here, factoring is a new skill, one that is likely yet to be mastered.  When struggling through these problems, review your ways of factoring.  There are four you currently possess.  They are as follows.

  1. Factor out a constant term (monomial)
  2. If quadratic and a = 1, find the factors of c that add to b.
  3. If quadratic and a ≠ 1, Guess and Check
  4. Factor by Grouping (for four term expressions)

A few tips.

  1. Difference of squares are factorable
    1. x4 – 81
  2. If the leading coefficient is negative, factor it out first as a monomial. It will make your methods of factoring work more smoothly.
    1. x2 – 2x – 1

 

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